Theorem mexval 33071

Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mexval.k	⊢ 𝐾 = (mTC‘𝑇)
mexval.e	⊢ 𝐸 = (mEx‘𝑇)
mexval.r	⊢ 𝑅 = (mREx‘𝑇)

Assertion

Ref	Expression
mexval	⊢ 𝐸 = (𝐾 × 𝑅)

Proof of Theorem mexval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mexval.e	. 2 ⊢ 𝐸 = (mEx‘𝑇)
2		fveq2 6699	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
3		mexval.k	. . . . . 6 ⊢ 𝐾 = (mTC‘𝑇)
4	2, 3	eqtr4di 2793	. . . . 5 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
5		fveq2 6699	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
6		mexval.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
7	5, 6	eqtr4di 2793	. . . . 5 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
8	4, 7	xpeq12d 5571	. . . 4 ⊢ (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅))
9		df-mex 33056	. . . 4 ⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
10		fvex 6712	. . . . 5 ⊢ (mTC‘𝑡) ∈ V
11		fvex 6712	. . . . 5 ⊢ (mREx‘𝑡) ∈ V
12	10, 11	xpex 7520	. . . 4 ⊢ ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V
13	8, 9, 12	fvmpt3i 6805	. . 3 ⊢ (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
14		xp0 6005	. . . . 5 ⊢ (𝐾 × ∅) = ∅
15	14	eqcomi 2749	. . . 4 ⊢ ∅ = (𝐾 × ∅)
16		fvprc 6691	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅)
17		fvprc 6691	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mREx‘𝑇) = ∅)
18	6, 17	syl5eq 2787	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
19	18	xpeq2d 5570	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅))
20	15, 16, 19	3eqtr4a 2801	. . 3 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
21	13, 20	pm2.61i 185	. 2 ⊢ (mEx‘𝑇) = (𝐾 × 𝑅)
22	1, 21	eqtri 2763	1 ⊢ 𝐸 = (𝐾 × 𝑅)

Syntax hints:  ¬ wn 3   = wceq 1543   ∈ wcel 2115  Vcvv 3401  ∅c0 4227   × cxp 5538  ‘cfv 6362  mTCcmtc 33033  mRExcmrex 33035  mExcmex 33036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2021  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2163  ax-12 2180  ax-ext 2712  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7505

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2076  df-mo 2542  df-eu 2572  df-clab 2719  df-cleq 2732  df-clel 2813  df-nfc 2883  df-ne 2937  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3403  df-sbc 3687  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-br 5044  df-opab 5106  df-mpt 5125  df-id 5444  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-iota 6320  df-fun 6364  df-fv 6370  df-mex 33056

This theorem is referenced by:  mexval2  33072  msubff  33099  msubco  33100  msubff1  33125  mvhf  33127  msubvrs  33129